Derivatives: Fundamental Concepts and Rules

This page delves into the core concepts of derivatives, presenting a comprehensive overview of derivative rules for various functions and operations.

Basic Derivative Rules

The page begins with a table of derivative rules for common functions, providing a quick reference for students. These rules form the foundation for more complex derivative calculations.

Highlight: Key derivative rules include:

• Constant function: f(x) = c, f'(x) = 0
• Linear function: f(x) = ax + b, f'(x) = a
• Power function: f(x) = x^n, f'(x) = nx^(n-1)
• Square root function: f(x) = √x, f'(x) = 1/(2√x)

Derivative Operations on Functions

The guide then explores how derivatives behave under various operations, such as addition, multiplication, and composition of functions.

Example: For the sum of functions (u + v), the derivative is u' + v'.

For the product of functions (u × v), the derivative is u'v + uv'.

These rules are essential for tackling more complex functions and real-world applications of calculus.

Special Derivative Rules

The page also covers some special derivative rules, including the chain rule and the quotient rule.

Vocabulary: The chain rule is used for composite functions, while the quotient rule is applied to ratios of functions.

Understanding these rules allows students to differentiate a wide range of functions encountered in advanced mathematics and physics.

Practical Applications

The importance of derivatives in various fields is emphasized, highlighting their role in optimization, rate of change problems, and physics applications.

Highlight: Derivatives are crucial in fields such as economics (marginal analysis), physics (velocity and acceleration), and engineering (optimization problems).

This page provides a solid foundation for students to tackle more advanced derivative problems and applications in subsequent lessons.

Derivative Examples and Problem-Solving Techniques

This page focuses on applying derivative rules to solve various types of problems, providing step-by-step examples to reinforce understanding.

Sum and Difference Rule Example

The page begins with an example demonstrating the application of the sum rule for derivatives.

Example: Find the derivative of f(x) = x² + √x

Solution: f'(x) = 2x + 1/(2√x)

This example illustrates how to break down a function into its components and apply the appropriate derivative rules.

Constant Multiple Rule Example

The guide then shows how to use the constant multiple rule in conjunction with other derivative rules.

Example: Find the derivative of f(x) = 5x³

Solution: f'(x) = 5 × 3x² = 15x²

This example demonstrates how to handle coefficients when differentiating power functions.

Product Rule Application

A more complex example involving the product rule is presented, showing how to differentiate the product of two functions.

Example: Find the derivative of f(x) = (x² + 1) × √x

Solution: f'(x) = (2x × √x) + (x² + 1) × (1/(2√x)) = 2x√x + (x² + 1)/(2√x)

This problem illustrates the step-by-step application of the product rule and the importance of careful algebraic manipulation.

Quotient Rule Demonstration

The page includes an example of using the quotient rule to differentiate a rational function.

Example: Find the derivative of f(x) = (2x - 3)/(3x + 1)

Solution: f'(x) = ((3x + 1) × 2 - (2x - 3) × 3) / (3x + 1)² = (6x + 2 - 6x + 9) / (3x + 1)² = 11 / (3x + 1)²

This example showcases the application of the quotient rule and the subsequent algebraic simplification.

Chain Rule Application

The final example on this page demonstrates the use of the chain rule for composite functions.

Example: Find the derivative of f(x) = √(3x + 1)

Solution: f'(x) = 1/(2√(3x + 1)) × 3 = 3/(2√(3x + 1))

This problem illustrates how to handle nested functions using the chain rule, a crucial skill for advanced calculus problems.